Step |
Hyp |
Ref |
Expression |
1 |
|
supicc.1 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
2 |
|
supicc.2 |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
3 |
|
supicc.3 |
⊢ ( 𝜑 → 𝐴 ⊆ ( 𝐵 [,] 𝐶 ) ) |
4 |
|
supicc.4 |
⊢ ( 𝜑 → 𝐴 ≠ ∅ ) |
5 |
|
supiccub.1 |
⊢ ( 𝜑 → 𝐷 ∈ 𝐴 ) |
6 |
|
supicclub2.1 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → 𝑧 ≤ 𝐷 ) |
7 |
|
iccssxr |
⊢ ( 𝐵 [,] 𝐶 ) ⊆ ℝ* |
8 |
1 2 3 4
|
supicc |
⊢ ( 𝜑 → sup ( 𝐴 , ℝ , < ) ∈ ( 𝐵 [,] 𝐶 ) ) |
9 |
7 8
|
sselid |
⊢ ( 𝜑 → sup ( 𝐴 , ℝ , < ) ∈ ℝ* ) |
10 |
3 7
|
sstrdi |
⊢ ( 𝜑 → 𝐴 ⊆ ℝ* ) |
11 |
10 5
|
sseldd |
⊢ ( 𝜑 → 𝐷 ∈ ℝ* ) |
12 |
10
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → 𝑧 ∈ ℝ* ) |
13 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → 𝐷 ∈ ℝ* ) |
14 |
12 13
|
xrlenltd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ( 𝑧 ≤ 𝐷 ↔ ¬ 𝐷 < 𝑧 ) ) |
15 |
6 14
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ¬ 𝐷 < 𝑧 ) |
16 |
15
|
nrexdv |
⊢ ( 𝜑 → ¬ ∃ 𝑧 ∈ 𝐴 𝐷 < 𝑧 ) |
17 |
1 2 3 4 5
|
supicclub |
⊢ ( 𝜑 → ( 𝐷 < sup ( 𝐴 , ℝ , < ) ↔ ∃ 𝑧 ∈ 𝐴 𝐷 < 𝑧 ) ) |
18 |
16 17
|
mtbird |
⊢ ( 𝜑 → ¬ 𝐷 < sup ( 𝐴 , ℝ , < ) ) |
19 |
9 11 18
|
xrnltled |
⊢ ( 𝜑 → sup ( 𝐴 , ℝ , < ) ≤ 𝐷 ) |